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Homework Help: Linear Transformations hints

  1. Feb 5, 2010 #1
    1. The problem statement, all variables and given/known data
    Let u = (1,2), v = (3,1) and T: [tex]R^{2}\rightarrow R[/tex] be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.)


    2. Relevant equations
    T(x)= b


    3. The attempt at a solution
    I don't really know where to start. I know I have to write (-3,4) as a linear combination of u and v, but what do I do from there?
     
  2. jcsd
  3. Feb 5, 2010 #2

    radou

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    Homework Helper

    The vectors u and v form a basis for R^2, since they're linearly independent (and there is two of them). So, you can write any vector in R^2 as their linear combination, with some coefficients α, β, so, for xome vector x (in your case (-3, 4) ), you have x = α u + β v. Now, what does T(x) equal to?
     
  4. Feb 5, 2010 #3
    So I determined α = 3 and β = -2 using your method. I was then able to solve for T(-3,4) = 2.

    Is this correct?
     
  5. Feb 5, 2010 #4

    radou

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    Yes, it is correct.
     
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